{
  "id": "cond-mat/0204243",
  "version": "v2",
  "published": "2002-04-11T10:12:28.000Z",
  "updated": "2002-04-13T10:50:16.000Z",
  "title": "Renormalization group approach to an Abelian sandpile model on planar lattices",
  "authors": [
    "Chai-Yu Lin",
    "Chin-Kun Hu"
  ],
  "comment": "29 pages, 6 figures",
  "doi": "10.1103/PhysRevE.66.021307",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "One important step in the renormalization group (RG) approach to a lattice sandpile model is the exact enumeration of all possible toppling processes of sandpile dynamics inside a cell for RG transformations. Here we propose a computer algorithm to carry out such exact enumeration for cells of planar lattices in RG approach to Bak-Tang-Wiesenfeld sandpile model [Phys. Rev. Lett. {\\bf 59}, 381 (1987)] and consider both the reduced-high RG equations proposed by Pietronero, Vespignani, and Zapperi (PVZ) [Phys. Rev. Lett. {\\bf 72}, 1690 (1994)] and the real-height RG equations proposed by Ivashkevich [Phys. Rev. Lett. {\\bf 76}, 3368 (1996)]. Using this algorithm we are able to carry out RG transformations more quickly with large cell size, e.g. $3 \\times 3$ cell for the square (sq) lattice in PVZ RG equations, which is the largest cell size at the present, and find some mistakes in a previous paper [Phys. Rev. E {\\bf 51}, 1711 (1995)]. For sq and plane triangular (pt) lattices, we obtain the only attractive fixed point for each lattice and calculate the avalanche exponent $\\tau$ and the dynamical exponent $z$. Our results suggest that the increase of the cell size in the PVZ RG transformation does not lead to more accurate results. The implication of such result is discussed.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2002-04-13T10:50:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "renormalization group approach",
      "abelian sandpile model",
      "planar lattices",
      "rg transformation",
      "exact enumeration"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "APS",
      "journal": "Phys. Rev. E"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}